Hilton-Milner Theorem for the $r$-independent sets in a union of cliques

TL;DR

This paper extends the Hilton-Milner theorem to $r$-independent sets in unions of cliques, providing maximum intersecting family characterizations and confirming a conjecture for a specific class of graphs called depth-two claws.

Contribution

It introduces a Hilton-Milner type theorem for $r$-independent sets in unions of cliques and applies it to prove the Holroyd--Talbot conjecture for depth-two claws.

Findings

Determined maximum intersecting families of $r$-independent sets in unions of cliques.

Established a tight upper bound for sums of sizes of cross intersecting families.

Confirmed the Holroyd--Talbot conjecture for all $r$ in depth-two claws graphs.

Abstract

We give a Hilton-Milner Theorem for the $r$-independent sets in the graph that is the union of copies of $K_k$. That is, we determine the maximum intersecting families of $r$-independent sets in this graph, subject to the condition that the sets in a family do not all share a common element. As a by-product, we also find a tight upper bound for the sum of sizes of a pair of cross intersecting families made up of the same objects. We apply our theorem to find the largest intersecting family of $r$-independent sets in a family of graphs called ``depth-two claws". This confirms the Holroyd--Talbot conjecture for depth-two claws, extending previous results on these graphs (which covered cases where $r$ was relatively small compared to the number of vertices) to all possible values of $r$.